Step | Hyp | Ref
| Expression |
1 | | frlmsslsp.y |
. . . . 5
⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
2 | 1 | frlmlmod 20093 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ LMod) |
3 | 2 | 3adant3 1081 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ LMod) |
4 | | eqid 2622 |
. . . 4
⊢
(LSubSp‘𝑌) =
(LSubSp‘𝑌) |
5 | | frlmsslsp.b |
. . . 4
⊢ 𝐵 = (Base‘𝑌) |
6 | | frlmsslsp.z |
. . . 4
⊢ 0 =
(0g‘𝑅) |
7 | | frlmsslsp.c |
. . . 4
⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} |
8 | 1, 4, 5, 6, 7 | frlmsslss2 20114 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝑌)) |
9 | | frlmsslsp.u |
. . . . . . . . . 10
⊢ 𝑈 = (𝑅 unitVec 𝐼) |
10 | 9, 1, 5 | uvcff 20130 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵) |
11 | 10 | 3adant3 1081 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈:𝐼⟶𝐵) |
12 | 11 | adantr 481 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑈:𝐼⟶𝐵) |
13 | | simp3 1063 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ 𝐼) |
14 | 13 | sselda 3603 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐼) |
15 | 12, 14 | ffvelrnd 6360 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐵) |
16 | | simpl2 1065 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐼 ∈ 𝑉) |
17 | | eqid 2622 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) |
18 | 1, 17, 5 | frlmbasf 20104 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝑦) ∈ 𝐵) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
19 | 16, 15, 18 | syl2anc 693 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅)) |
20 | | simpll1 1100 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring) |
21 | | simpll2 1101 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉) |
22 | 14 | adantr 481 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ∈ 𝐼) |
23 | | eldifi 3732 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼) |
24 | 23 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼) |
25 | | disjdif 4040 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ |
26 | | disjne 4022 |
. . . . . . . . . 10
⊢ (((𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
27 | 25, 26 | mp3an1 1411 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
28 | 27 | adantll 750 |
. . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥) |
29 | 9, 20, 21, 22, 24, 28, 6 | uvcvv0 20129 |
. . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝑦)‘𝑥) = 0 ) |
30 | 19, 29 | suppss 7325 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽) |
31 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = (𝑈‘𝑦) → (𝑥 supp 0 ) = ((𝑈‘𝑦) supp 0 )) |
32 | 31 | sseq1d 3632 |
. . . . . . 7
⊢ (𝑥 = (𝑈‘𝑦) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
33 | 32, 7 | elrab2 3366 |
. . . . . 6
⊢ ((𝑈‘𝑦) ∈ 𝐶 ↔ ((𝑈‘𝑦) ∈ 𝐵 ∧ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)) |
34 | 15, 30, 33 | sylanbrc 698 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐶) |
35 | 34 | ralrimiva 2966 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶) |
36 | | ffn 6045 |
. . . . . . 7
⊢ (𝑈:𝐼⟶𝐵 → 𝑈 Fn 𝐼) |
37 | 11, 36 | syl 17 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈 Fn 𝐼) |
38 | | fnfun 5988 |
. . . . . 6
⊢ (𝑈 Fn 𝐼 → Fun 𝑈) |
39 | 37, 38 | syl 17 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Fun 𝑈) |
40 | | fndm 5990 |
. . . . . . 7
⊢ (𝑈 Fn 𝐼 → dom 𝑈 = 𝐼) |
41 | 37, 40 | syl 17 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → dom 𝑈 = 𝐼) |
42 | 13, 41 | sseqtr4d 3642 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ dom 𝑈) |
43 | | funimass4 6247 |
. . . . 5
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
44 | 39, 42, 43 | syl2anc 693 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)) |
45 | 35, 44 | mpbird 247 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐶) |
46 | | frlmsslsp.k |
. . . 4
⊢ 𝐾 = (LSpan‘𝑌) |
47 | 4, 46 | lspssp 18988 |
. . 3
⊢ ((𝑌 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝑌) ∧ (𝑈 “ 𝐽) ⊆ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
48 | 3, 8, 45, 47 | syl3anc 1326 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶) |
49 | | simpl1 1064 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑅 ∈ Ring) |
50 | | simpl2 1065 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝐼 ∈ 𝑉) |
51 | | ssrab2 3687 |
. . . . . . . . 9
⊢ {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⊆ 𝐵 |
52 | 7, 51 | eqsstri 3635 |
. . . . . . . 8
⊢ 𝐶 ⊆ 𝐵 |
53 | 52 | a1i 11 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ⊆ 𝐵) |
54 | 53 | sselda 3603 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐵) |
55 | | eqid 2622 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌) |
56 | 9, 1, 5, 55 | uvcresum 20132 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈))) |
57 | 49, 50, 54, 56 | syl3anc 1326 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 = (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈))) |
58 | | eqid 2622 |
. . . . . 6
⊢
(0g‘𝑌) = (0g‘𝑌) |
59 | | lmodabl 18910 |
. . . . . . . 8
⊢ (𝑌 ∈ LMod → 𝑌 ∈ Abel) |
60 | 3, 59 | syl 17 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ Abel) |
61 | 60 | adantr 481 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑌 ∈ Abel) |
62 | | imassrn 5477 |
. . . . . . . . . 10
⊢ (𝑈 “ 𝐽) ⊆ ran 𝑈 |
63 | | frn 6053 |
. . . . . . . . . . 11
⊢ (𝑈:𝐼⟶𝐵 → ran 𝑈 ⊆ 𝐵) |
64 | 11, 63 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ran 𝑈 ⊆ 𝐵) |
65 | 62, 64 | syl5ss 3614 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐵) |
66 | 5, 4, 46 | lspcl 18976 |
. . . . . . . . 9
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
67 | 3, 65, 66 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
68 | 4 | lsssubg 18957 |
. . . . . . . 8
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
69 | 3, 67, 68 | syl2anc 693 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
70 | 69 | adantr 481 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌)) |
71 | 1, 17, 5 | frlmbasf 20104 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
72 | 71 | 3ad2antl2 1224 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
73 | | ffn 6045 |
. . . . . . . . . 10
⊢ (𝑦:𝐼⟶(Base‘𝑅) → 𝑦 Fn 𝐼) |
74 | 72, 73 | syl 17 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦 Fn 𝐼) |
75 | 37 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈 Fn 𝐼) |
76 | | simpl2 1065 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝐼 ∈ 𝑉) |
77 | | inidm 3822 |
. . . . . . . . 9
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
78 | 74, 75, 76, 76, 77 | offn 6908 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
79 | 54, 78 | syldan 487 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
80 | 54, 74 | syldan 487 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
81 | 80 | adantrr 753 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑦 Fn 𝐼) |
82 | 37 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑈 Fn 𝐼) |
83 | | simpl2 1065 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝐼 ∈ 𝑉) |
84 | | simprr 796 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑧 ∈ 𝐼) |
85 | | fnfvof 6911 |
. . . . . . . . . . 11
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
86 | 81, 82, 83, 84, 85 | syl22anc 1327 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
87 | 3 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑌 ∈ LMod) |
88 | 67 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
89 | 52 | sseli 3599 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) |
90 | 89, 72 | sylan2 491 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦:𝐼⟶(Base‘𝑅)) |
91 | 90 | adantrr 753 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑦:𝐼⟶(Base‘𝑅)) |
92 | 13 | sselda 3603 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
93 | 92 | adantrl 752 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑧 ∈ 𝐼) |
94 | 91, 93 | ffvelrnd 6360 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘𝑅)) |
95 | 1 | frlmsca 20097 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝑌)) |
96 | 95 | 3adant3 1081 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑅 = (Scalar‘𝑌)) |
97 | 96 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
98 | 97 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
99 | 94, 98 | eleqtrd 2703 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌))) |
100 | 5, 46 | lspssid 18985 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
101 | 3, 65, 100 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
102 | 101 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽))) |
103 | | funfvima2 6493 |
. . . . . . . . . . . . . . . . . 18
⊢ ((Fun
𝑈 ∧ 𝐽 ⊆ dom 𝑈) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
104 | 39, 42, 103 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))) |
105 | 104 | imp 445 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
106 | 105 | adantrl 752 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)) |
107 | 102, 106 | sseldd 3604 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
108 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
109 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
110 | 108, 55, 109, 4 | lssvscl 18955 |
. . . . . . . . . . . . . 14
⊢ (((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) ∧ ((𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
111 | 87, 88, 99, 107, 110 | syl22anc 1327 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
112 | 111 | anassrs 680 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
113 | 112 | adantlrr 757 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
114 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
115 | 114 | adantrr 753 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
116 | 115 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶)) |
117 | | simplrr 801 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼) |
118 | | simpr 477 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ¬ 𝑧 ∈ 𝐽) |
119 | 117, 118 | eldifd 3585 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ (𝐼 ∖ 𝐽)) |
120 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → (𝑥 supp 0 ) = (𝑦 supp 0 )) |
121 | 120 | sseq1d 3632 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑦 → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑦 supp 0 ) ⊆ 𝐽)) |
122 | 121, 7 | elrab2 3366 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 supp 0 ) ⊆ 𝐽)) |
123 | 122 | simprbi 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ 𝐶 → (𝑦 supp 0 ) ⊆ 𝐽) |
124 | 123 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ⊆ 𝐽) |
125 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . 19
⊢
(0g‘𝑅) ∈ V |
126 | 6, 125 | eqeltri 2697 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
V |
127 | 126 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 0 ∈ V) |
128 | 90, 124, 50, 127 | suppssr 7326 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (𝑦‘𝑧) = 0 ) |
129 | 116, 119,
128 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = 0 ) |
130 | 96 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (0g‘𝑅) =
(0g‘(Scalar‘𝑌))) |
131 | 6, 130 | syl5eq 2668 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 0 =
(0g‘(Scalar‘𝑌))) |
132 | 131 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 0 =
(0g‘(Scalar‘𝑌))) |
133 | 129, 132 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = (0g‘(Scalar‘𝑌))) |
134 | 133 | oveq1d 6665 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) =
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧))) |
135 | 3 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑌 ∈ LMod) |
136 | 11 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐼) → (𝑈‘𝑧) ∈ 𝐵) |
137 | 136 | adantrl 752 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → (𝑈‘𝑧) ∈ 𝐵) |
138 | 137 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ 𝐵) |
139 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(0g‘(Scalar‘𝑌)) =
(0g‘(Scalar‘𝑌)) |
140 | 5, 108, 55, 139, 58 | lmod0vs 18896 |
. . . . . . . . . . . . . 14
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑧) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
141 | 135, 138,
140 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
142 | 134, 141 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌)) |
143 | 67 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) |
144 | 58, 4 | lss0cl 18947 |
. . . . . . . . . . . . 13
⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
145 | 135, 143,
144 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽))) |
146 | 142, 145 | eqeltrd 2701 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
147 | 113, 146 | pm2.61dan 832 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦‘𝑧)( ·𝑠
‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
148 | 86, 147 | eqeltrd 2701 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
149 | 148 | expr 643 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑧 ∈ 𝐼 → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) |
150 | 149 | ralrimiv 2965 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ∀𝑧 ∈ 𝐼 ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))) |
151 | | ffnfv 6388 |
. . . . . . 7
⊢ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽)) ↔ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 ∧ ∀𝑧 ∈ 𝐼 ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) |
152 | 79, 150, 151 | sylanbrc 698 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽))) |
153 | 1, 6, 5 | frlmbasfsupp 20102 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 ) |
154 | 153 | fsuppimpd 8282 |
. . . . . . . . 9
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈
Fin) |
155 | 50, 54, 154 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ∈
Fin) |
156 | | dffn2 6047 |
. . . . . . . . . . 11
⊢ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 ↔ (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
157 | 78, 156 | sylib 208 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈):𝐼⟶V) |
158 | 74 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑦 Fn 𝐼) |
159 | 37 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑈 Fn 𝐼) |
160 | | simpll2 1101 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝐼 ∈ 𝑉) |
161 | | eldifi 3732 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 )) → 𝑥 ∈ 𝐼) |
162 | 161 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑥 ∈ 𝐼) |
163 | | fnfvof 6911 |
. . . . . . . . . . . 12
⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
164 | 158, 159,
160, 162, 163 | syl22anc 1327 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
165 | | ssid 3624 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ) |
166 | 165 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 )) |
167 | 126 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 0 ∈ V) |
168 | 72, 166, 76, 167 | suppssr 7326 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = 0 ) |
169 | 131 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 0 =
(0g‘(Scalar‘𝑌))) |
170 | 168, 169 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = (0g‘(Scalar‘𝑌))) |
171 | 170 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦‘𝑥)( ·𝑠
‘𝑌)(𝑈‘𝑥)) =
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥))) |
172 | 3 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑌 ∈ LMod) |
173 | 11 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈:𝐼⟶𝐵) |
174 | | ffvelrn 6357 |
. . . . . . . . . . . . 13
⊢ ((𝑈:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) ∈ 𝐵) |
175 | 173, 161,
174 | syl2an 494 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑈‘𝑥) ∈ 𝐵) |
176 | 5, 108, 55, 139, 58 | lmod0vs 18896 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑥) ∈ 𝐵) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
177 | 172, 175,
176 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) →
((0g‘(Scalar‘𝑌))( ·𝑠
‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌)) |
178 | 164, 171,
177 | 3eqtrd 2660 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)‘𝑥) = (0g‘𝑌)) |
179 | 157, 178 | suppss 7325 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
180 | 54, 179 | syldan 487 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) |
181 | | ssfi 8180 |
. . . . . . . 8
⊢ (((𝑦 supp 0 ) ∈ Fin ∧ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 )) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin) |
182 | 155, 180,
181 | syl2anc 693 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin) |
183 | | simp2 1062 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐼 ∈ 𝑉) |
184 | 1, 17, 5 | frlmbasmap 20103 |
. . . . . . . . . . . 12
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
185 | 183, 89, 184 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
186 | | elmapfn 7880 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚
𝐼) → 𝑦 Fn 𝐼) |
187 | 185, 186 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼) |
188 | 11 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈:𝐼⟶𝐵) |
189 | 188, 36 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈 Fn 𝐼) |
190 | 187, 189,
50, 50, 77 | offn 6908 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼) |
191 | | fnfun 5988 |
. . . . . . . . 9
⊢ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) Fn 𝐼 → Fun (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) |
192 | 190, 191 | syl 17 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → Fun (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) |
193 | | ovexd 6680 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) ∈ V) |
194 | | fvex 6201 |
. . . . . . . . 9
⊢
(0g‘𝑌) ∈ V |
195 | 194 | a1i 11 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (0g‘𝑌) ∈ V) |
196 | | funisfsupp 8280 |
. . . . . . . 8
⊢ ((Fun
(𝑦
∘𝑓 ( ·𝑠 ‘𝑌)𝑈) ∧ (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) ∈ V ∧ (0g‘𝑌) ∈ V) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin)) |
197 | 192, 193,
195, 196 | syl3anc 1326 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin)) |
198 | 182, 197 | mpbird 247 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌)) |
199 | 58, 61, 50, 70, 152, 198 | gsumsubgcl 18320 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑌 Σg (𝑦 ∘𝑓 (
·𝑠 ‘𝑌)𝑈)) ∈ (𝐾‘(𝑈 “ 𝐽))) |
200 | 57, 199 | eqeltrd 2701 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (𝐾‘(𝑈 “ 𝐽))) |
201 | 200 | ex 450 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑦 ∈ 𝐶 → 𝑦 ∈ (𝐾‘(𝑈 “ 𝐽)))) |
202 | 201 | ssrdv 3609 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ⊆ (𝐾‘(𝑈 “ 𝐽))) |
203 | 48, 202 | eqssd 3620 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) |